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Recently, Dong (Aequ Math 86:269-277, ) has proved the generalized stability of the functional equation $${\|f(x + y)\| = \|f(x) + f(y)\|}$$ under the assumption that X, the domain of f, is an Abelian group. In this paper, we prove a generalization of this result by removing the commutativity assumption of X. [ABSTRACT FROM AUTHOR]

The diagonalization of a two-variable functional equation (related to competition) leads to the iterative equation It was shown in Kahlig (Appl Math 39:293-303, ) that if a function $${f:{\mathbb{R}}\rightarrow {\mathbb{R}}}$$ , such that f(0) = 0, satisfies this equation for all $${x\in (-1,1),}$$ and is twice differentiable at the point 0, then $${f=\tanh \circ (p\,\tan ^{-1}) }$$ for some real p. In this paper we prove the following stronger result. A function $${f:{\mathbb{R}} \rightarrow {\mathbb{R}},\;f(0) =0}$$ , differentiable at the point 0, satisfies this functional equation if, and only if, there is a real p such that $${f=\tanh \circ (p\,\tan ^{-1})}$$ . We also show that the assumption of the differentiability of f at 0 cannot be replaced by the continuity of f. The corresponding result for the iterative equation coming from a three- respectively four-variable competition equation is also proved. Our conjecture is that analogous results hold true for the diagonalization of any n-variable competition equation $${(n=5, 6, 7, \ldots)}$$ . [ABSTRACT FROM AUTHOR]

This paper treats a functional equation of d'Alembert, namely f( x + y) − f( x − y) = g( x) h( y), in the setting of groups and monoids that need not be abelian. Of the several possibilities for generalizing x − y to the non-abelian case, here we replace it by σ( y) x where σ is an involutive automorphism, and find the complex-valued solutions ( f, g, h) of f( xy) − f( σ( y) x) = g( x) h( y) in terms of multiplicative and additive functions. [ABSTRACT FROM AUTHOR]

The functional equation related to competition ([]) for y = cx with a fixed c > 0, leads to the equation The case c = 1 (a first order iterative functional equation) was treated in []. In this paper we consider the case c ≠ 1 (when the equation is of the second order). We show that a function $${f:\mathbb{R} \rightarrow \mathbb{R},\,f\left( 0\right) =0}$$ , differentiable at the point 0 satisfies this functional equation iff there is a real p such that $${f=\tanh \circ \left( p\tan ^{-1} \right) }$$ which extends the main result of []. [ABSTRACT FROM AUTHOR]

In this paper we determine the general solution of some special cases of the generalized Hosszú functional equation on intervals [0,1] and (0,1). [ABSTRACT FROM AUTHOR]

Given rational vector spaces V, W a mapping $${f \colon V \to W}$$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f of degree i such that $${f = \sum_{i = 0}^n f_i}$$ . Homogeneous generalized polynomials f of degree i are mappings of the form $${f_i (x) = f_i^*(x, x, \ldots , x)}$$ with $${f_i^* \colon V^i \to W i}$$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f or $${f_n + f_{n - 1}}$$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset $${L \subseteq \{0, 1, \ldots, n\}}$$ such that $${n \in L}$$ , a method is described to find (many) functional equations, such that their general solution is given by $${\sum_{i \in L} f_i}$$ . For the cases $${L = \{n\}}$$ and $${L = \{n - 1, n\}}$$ additional equations are given. [ABSTRACT FROM AUTHOR]

In this paper we prove that if a generalized polynomial function f satisfies the condition f( x) f( y) = 0 for all solutions of the equation x + y = 1, then f is identically equal to 0. [ABSTRACT FROM AUTHOR]